= Inverse conjugation and adjoint antirepresentations
{title2=$F(gh)=F(h)F(g),\quad F(g)=\operatorname{Ad}_{g^{-1}}$}
With <Adjoint representation> convention $\operatorname{ad}_X(Y)=[X,Y]$, the <matrix exponential> identity is $e^{-X}Ye^X=e^{-\operatorname{ad}_X}Y$. Inverse conjugation reverses product order, so it is a right action or antirepresentation, whereas $\operatorname{Ad}_g(Y)=gYg^{-1}$ is an ordinary left <group representation>. The first-order terms $Y\mp[X,Y]$ detect a sign mismatch immediately. Negating every adjoint generator without reversing the Lie bracket does not give a new Lie-algebra homomorphism.
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